Kepler and the congruences for a vertical teaching activity

“A plane figure is said to be regular if it has all its sides and all its outward-facing angles equal to one another”. “Some of these are primary and basic, not extending beyond their boundaries, and it is to these that the previous definition prop­ erly applies; others are augmented, as it were extending beyond their sides, and if two non-neighboring sides of one of the basic figures are produced they meet: these are called Stars”. With these two definitions, Johannes Kepler begins the first book of his “Harmonices Mundi” of 1618. Divided into five books, Kepler dedicates the first two to regular polygons, tilings and polyhedra. Starting from the Middle Ages and throughout the Renaissance, star figures had been widely used in art and architecture but Kepler was the first to give them the status of polygon; the obvious consequence is that he can use them in congruences on the plane. Congruences are today known by the terms tessellation or tiling and the best known of them, especially in in the field of teaching, are the regular and periodic ones. There are others that are much more complex and sophisticated and Kepler, probably for the first time, presented a study that we could call "methodological" on the possible tessellations of the plane and space. We presented Kepler's method in a vertical path on the study of polygons and tessellations. We presented part of this activity at courses for students and teachers: the "Orienta-life" project proposed by the Campania regional school office and the project "Competence-oriented mathematical activities: towards new horizons". Our educational activity on the tessellations of the plane retraces, for the construction of the star figures, some stages of the evolution of the concept from a historical point of view. The activities involve manipulation of concrete material; identification of angles (internal and external); angle measurements; exploration of concave polygons. Furthermore, the interdisciplinary aspect promotes the learning of mathematical concepts. In general, the use of the history of mathematics in teaching is justified by various factors and the materials used are, mainly, primary sources and historical documents; but the story also concerns the way in which a concept evolved, the different approaches of mathematicians of the past, their difficulties, their mathematical creativity up to the formalization phase. Skemp (The psychology of learning mathematics. UK: Penguin, 1969) states that a purely logical approach only provides the final product of mathematical discovery and does not generate in learning the processes by which mathematical discoveries are made. When mathematics is presented as a "complete" discipline to students, they have difficulty in identifying the cognitive roots of concepts in the magma of processes, concepts and rules: implementing educational activities in the form of workshops favored operations and at the same time dialogue and reflection. We have therefore chosen an approach consistent with the way of proceeding of mathematics, respecting the experimental character of the discipline; we have shown that it is possible to resume with the students the path of research, exploration, discovery and construction that is the basis of a mathematical discovery. Here we want to present the structure of our activity, describing the phases already implemented in the classroom, highlighting connections with other disciplines, and future developments.