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.