Seminar on architecture and geometry in the sketchbooks of Leonardo. At least two distinct uses of geometry can be found in Leonardo's sketchbook. The first involves geometry as the basis for structural stability. The point of departure for the investigation of this aspect is found in page 899v in the Codex Atlanticus in which Leonardo illustrates the design of a roof system assembled from loose parts. This kind of structure is similar to his design for a wooden bridge, a model of which is on display in the Leonardo Museum in Vinci. Topics examined were the precise role of geometry in supporting this structure (how, for example, the issue of geometrical shape contributes to structural stability), what mathematics were used by Leonardo in the discovery of this system, and what mathematics are required for find all possible Leonardo grids (there are more than one hundred of them). The intention is to shed light on Leonardo's process for discovering his system.

Detail of Leonardo's page 899v in the Codex Atlanticus

The second use of geometry in Leonardo's architecture is as an underlying organizational structure for the plans of churches, as shown in MS 2307, fol. 5v. During the seminar there was a discussion of what proportional systems are generated by such geometries; if there might be any significant symbolism connected to the use of a particular geometry; if Leonardo was drawing on a geometrical tradition used by contemporary architects.

The use of geometry at the seminar was studied through the use of traditional geometrical tools such as compass and straightedge and the building of models, as well as through modern technologies of and computer modeling and animation.

The construction. It is important to take the discussion from the realm of theory into the realm of practice. The construction of the dome allowed the theory to be tested. The construction was directed by Dutch artist Rinus Roelofs. He has worked with Leonardo's system of bar grids since 1989, and writes,

In 1989 I made a drawing of a net on a cube. The net consisted of 12 lines/elements and they were connected in a way that I recognised a couple of months later in the da Vinci drawings. What we see on the da Vinci drawings are some examples of roof constructions built with a lot of straight elements (see illustration of page 899v above). These drawings can be 'translated' into the following definition: On each element we define four points at some distance of each other'. So we get two points somewhere in the middle and two points somewhere near the ends. To make constructions with these elements we may only connect a middle point of one element to an end point of another one. Dome building is only one of the things you can do with the Leonardo grids. Next step is sphere building, and you can go on with cylinders, columns and even foldables.

The basic modules for constructing Leonardo's roof structures

Project participants at work constructing wooden models of the dome structures

One of the four wooden cupolas constructed during the two days of experimental construction

Diffusion of the results. The results seminar will be made available in several different media. Planned are an exhibit of the project results at the Leonardo Museum in Vinci (tentatively scheduled for September/October 2004); a "virtual", interactive, exhibit dedicated to Leonardo's architecture and geometry; an exhibit catalog including the papers by seminar participants (Kim Williams Books, 2004); the video documentary.

The participants.

Kim Williams of Fucecchio (Florence), Italy, is an American architect and scholar. She is the director of the biennial conference series "Nexus: Architecture and Mathematics" and editor-in-chief of the Nexus Network Journal, a peer-reviewed quarterly dedicated to studies in architecture and mathematics. She received her degree in Architectural Studies from the University of Texas in Austin, and is licensed as an architect in New York State. She is the author of Italian Pavements. Patterns in Space (Houston: Anchorage Press, 1997) about the role of decorated pavements in the history of Italian architecture. She edits the Nexus book series. She has published many articles in scholarly journals on the use of mathematical principles in architecture, including "The Sacred Cut Revisited: The Pavement of the Baptistery of San Giovanni, Florence" (Mathematical Intelligencer, 16, no. 2, Spring 1994, 18-24); "Michelangelo's Medici Chapel: The Cube, The Square and the Root-2 Rectangle" (Leonardo. Journal of the International Society of Arts, Sciences and Technology, 30, no. 2 (1997), 105-112); "Spirals and the Rosette in Architectural Ornament" (Applied Geometry. MAA Notes 53. Catherine Gorini, et al., eds. Mathematical Association of America, 2000). Her drawings have been displayed in both group and solo exhibits. She has participated in numerous international conferences, and regularly lectures on architecture and mathematics.

Biagio Di Carlo is an architect as well as a graphic designer and a musician. He received his degree in Architectural studies with honors in 1976 from the University of Architecture of Pescara; his thesis was published. His thesis advisors were Eduardo Vittoria , together with Giovanni Guazzo and Augusto Vitale. He teaches at the Art Institute of Pescara, and often collaborates with the architectural faculties of Pescara and Ascoli, by giving lectures, lessons and seminars in synergetic geometry, geodesic domes, tensegrities, quasi crystals and four dimensional polytopes. He is the author of the self-published book STRUTTURE GEODETICHE has published articles on synergetic geometry and geodesic domes in journals such as Bioarchitettura and L'Architettura naturale. Among his interests are architecture, molecular geometry, graphic arts and cartoon illustrations, Latin jazz and Brasilian music. More information about his activities is available on his website.

Sylvie Duvernoy of Florence, Italy, is an architect who graduated from Paris University in 1982. She later participated in the Ph.D. program of the Architecture School of University of Florence and was awarded the Italian degree of "Dottore di Ricerca" in 1998. She presently teaches architectural drawing at the engineering and architecture faculties of University of Florence. Since she began post-graduate studies, her research has mainly focused on the reciprocal influences between graphic mathematics and architecture. Architectural history shows that geometry and its related aesthetic symbolism were always present, hidden in architectural and urban design from antiquity to modern times. The type of application and the explicitness of their use vary according to historical periods. These relationships have always been expressed by the means of the drawing: the major and unavoidable tool of the design process. The results of her studies were published and communicated in several international meetings and reviews. In June 2002 she presented "Architecture and Mathematics in Roman Amphitheaters" at the Nexus 2002 conference in Obidos, Portugal. In addition to research and teaching, she maintains a private practice as an architect. After having worked for a few years in the Parisian office of an international Swiss architecture firm, she is now partner of an associate office in Florence, the design projects of which cover a wide range of design problems, from remodeling and restoration to new constructions, in Italy and abroad.

Christopher Glass of Camden, Maine, USA, is an architect with a one-person practice in coastal Maine. He attended Saint Albans School in Washington D.C., studied philosophy at Haverford College and architecture at Yale. He teaches an introductory architecture studio at Bowdoin College and is trying to cut back on professional work to spend more time playing with toys like the Pythagopod, about which he has written in an article for the Nexus Network Journal.

Vesna Petresinof Ljubljana, Slovenia, now living in London, is an architect. She earned her Ph.D. in temporal aspects of architectural composition at the Faculty of Architecture in Ljubljana. She has worked with Laurent-Paul Robert for Stephen Perrella's Hypersurface Systems and taught studios with Mark Goulthorpe and Tom Kovac. Since 2001, she continues her design and research with Laurent-Paul Robert as Rubedo. Her research interests include visual theory, time-based design processes, interactive architecture and hybrid space. Having graduated from the Musical Conservatory in Ljubljana, her projects also include musical theory and opera singing.

Mark Reynolds is a visual artist who works primarily in drawing, printmaking and mixed media. He received his Bachelor's and Master's Degrees in Art and Art Education at Towson University in Maryland. He was awarded the Andelot Fellowship to do post-graduate work in drawing and printmaking at the University of Delaware. For the past decade, Mr. Reynolds has been at work on an extensive body of drawings, paintings and prints that incorporate and explore the ancient science of sacred, or contemplative, geometry. He is widely exhibited, showing his work in group competitions and one person shows, especially in California. Mark's work is in corporate, public, and private collections. Mark is also a member of the California Society of Printmakers (six of his images can be found on their website by clicking on "Galleries" then scrolling down to Mark Reynolds under "Artist Member Porfolios), the Los Angeles Printmaking Society, and the Marin Arts Council.
A born teacher, Mr. Reynolds teaches sacred geometry, linear perspective, drawing, and printmaking to both graduate and undergraduate students in various departments at the Academy of Art College in San Francisco, California. He was voted Outstanding Educator of the Year by the students in 1992.
Additionally, Reynolds is a geometer, and his specialties in this field include doing geometric analyses of architecture, paintings, and design. He presented, "A New Geometric Analysis of the Pazzi Chapel", at The Nexus 2000 Conference in Ferrara, Italy, and published, "A Comparative Geometric Analysis of the Heights and Bases of The Great Pyramid of Khufu and The Pyramid of the Sun in Teotihuacan", in The Nexus Network Journal, vol. 1, no. 4. Mr. Reynolds is also contributing editor for the "Geometer's Angle" column in the Nexus Network Journal. He lives with his wife and family in Mill Valley, California.

Laurent-Paul Robert is a French/Swiss artist and computer animator based in London. After having worked as an artist, performer and musician, he has continued in 3D animation since 1994, becoming the lead art director and senior animator for Arxel Tribe. He also worked in architecture with Stephen Perrella's Hypersurface Systems, and has continued his design and research with Vesna Petresin as Rubedo since 2001; he has exhibited and published internationally. He is currently working as a consultant for clients such as Arup and dECOi

Rinus Roelofs of Hengelo, Netherlands, is an artist. He was born in 1954. After studying Applied Mathemathics at the Technical University of Enschede, he took a degree from the Enschede Art Academy with a specialization in sculpture. His commissions come largely from municipalities, institutions and companies in the Netherlands, but his work has been exhibited further afield, including in Rome as part of the Escher Centennial celebrations in 1998. More information about his activities is available on his website.

João Pedro Xavier of Oporto, Portugal, is an architect and geometry teacher. Her eceived his degree in Architecture from the Faculty of Architecture of the University of Porto (FAUP) and is licensed as an architect at the College of Architects in Porto since 1986. He won academic prizes, including the"Prémio Florêncio de Carvalho" and "Prémio Engº António de Almeida". He worked in Álvaro Siza's office from 1986 to 1999. At the same time he established his own practice as an architect. He has participated in several exhibitions, courses and seminars. One of his latest projects was the Exhibition "Matemática Viva" (an interactive exhibition on mathematics at the Pavilhão do Conhecimento in Lisbon, organized by the Association ATRACTOR), for which he also conceived all the modules on perspective. He has been teaching geometry since 1985 at the Architecture School of Cooperativa Árvore in Porto, the Fine Arts School of Porto and at FAUP from 1991 onwards. He wrote Perspectiva, perspectiva acelerada e contraperspectiva (FAUP Publicações, 1997). Now he is preparing his Phd on the same subject, advised by Prof. Arch. Alexandre Alves Costa. Xavier has always been interested in the relationship between architecture and mathematics, especially geometry. He published several works and papers on the subject, made conferences and lectures and gave courses to high school teachers. He also collaborated with the Ministry of Education coordinating the team in charge of the elaboration of Descriptive Geometry curricula in Portugal. In June 2002 he presented "António Rodrigues, a Portuguese architect with a scientific inclination" at the Nexus 2002 conference in Obidos, Portugal.

The Leonardo project participants and their first cupola