La magia dei gruppi di simmetria. I Gruppi dei Magnifici Rosoni

During our meetings with the students of architecture we often felt the need, and the desire, of displaying some application that would motivate the teaching of mathematics in the schools of architecture, where it is often perceived only as a tool for the other disciplines – often considered more useful or noble. We thought that the problem could be solved with the identification of the “perfect” subject: short enough to keep readers interested, independent from any other, relatively easy to access, but in same time with a non-trivial and elegant mathematical content. The symmetry groups of the plane appeared to be such "perfect" subject. The idea was developed during some lessons of the course "Mathematical Methods for Architecture" held at the Faculty of Architecture of the University of Genoa. There was the need indeed to explain to a non expert audience the connection between the geometric structure of the canopy of a Romanesque or Gothic church, a particular decoration of the Arab or the floor of a Christian basilica and a refined sequence of arguments relating to the algebra of rigid transformations of the plane. In order not to frighten the reader, causing him to prematurely abandon a topic that could fascinate both the art lover and the person sensitive to the elegance of geometric reasoning, we will start with the finite groups of the plan renaming them as the "groups of magnificent roses." The book will be divided into two chapters. The first one will introduce the reader – in technical and mathematical terms – to the classification of rigid transformations of the plane, otherwise called isometries, up to the characterization of the finite symmetry groups. An extensive use of graphics will help the reader to understand the technical justifications that also explain the definition of discrete group. On the other hand, who appreciates the elegance of mathematical reasoning could perhaps find here what can be described as a sophisticated use of elementary concepts. In the second chapter, for each of the two finite groups of the plan we give various examples of their use in architecture and art, with the hope of involving the reader in trying to recognize the different ornaments whenever they encounter them.