Friezes and Mosaics

Abstract

This chapter discusses the classification of friezes and several concepts related to mosaics. The first section introduces the concept of operations that leave a frieze unchanged , using basic geometry and intuition. It also describes what will be the main steps of the classification theorem. Section 2.2 defines affine transformations and their matrix representation, and isometries. The highlight of this chapter is the classification theorem shown in Section 2.3. In less detail, the last section discusses mosaics. There is no advanced section to this chapter, the proof of the classification theorem being the most difficult element. Sections 2.1 and 2.4 can be covered in three hours of class. The tools are then purely geometric and the possibility of classification is made clear. If the classification theorem is the goal, four hours should be devoted to the first three sections. In all cases, the lecturer should bring copies of Figure 2.2 on transparencies to the classroom. Their use on a projector helps students to understand quickly the concept of symmetry. Only a basic knowledge of linear algebra and Euclidean geometry is required to understand this chapter. The proof of the classification theorem requires a familiarity with abstract reasoning. This subject offers several interesting directions for further study: aperiodic tilings (end of Section 2.4) is one such direction, while Exercises 13, 14, 15, and 16 present several others. Friezes and mosaics have been used in decoration for several millennia. The ancient world's Sumerian, Egyptian, and Mayan civilizations all used them to great effect. It would be a lie, however, to pretend that ancient mathematics developed the "tech-nology" behind the art. The formal mathematical study of tilings is relatively recent, having started no more than two centuries ago. The memoir of Bravais [1], a French physicist, is among the first scientific studies of the subject. Mathematics is able to provide a way to systematically classify the friezes and mosaics commonly seen in architecture and art. These classifications have allowed mathematicians to better understand the rules behind them and to create truly new patterns by breaking some of these rules.