Um estudo sobre os poliedros de Platão, Arquimedes, Catalan e Kepler-Poinsot e suas construções no GeoGebra

Abstract

This research aimed to investigate the pedagogical advantages and difficulties of using the GeoGebra software in the construction of several classes of polyhedrons during a continuing education course for mathematics teachers in the state network of Paraíba. The polyhedra studied during the course were those of Plato, Archimedes, Kepler-Poinsot and Catalan solids. We are guided by the following research question: How can the use of GeoGebra software help mathematics teachers in their pedagogical practice? The theoretical framework is based on Euclides' work, The Elements. We observe how he defined and approached the construction of regular polyhedra. The research is qualitative in nature, since it is based on the reading of Book XIII of Os Elementos de Euclides. The procedures were developed in three parts. In the first part, we approach a little of the history of geometry, highlighting the polyhedra and how this mathematical object was mentioned in the works The Elements, by Euclid, and in the Timaeus-Crítias, by Plato’s. In this first part we also present a little about the compositional form of Euclid's masterpiece. In the second part, we approach the difficulties in defining polyhedron, as well as the different conceptions of this mathematical object. We clarify the difference between Plato's polyhedra and regular polyhedra, which are concepts often used synonymously. We present the concept of dual polyhedra, explain the characteristics (number of faces, vertices and edges) of regular, Archimedean, Catalan and Kepler-Poinsot polyhedra. In the third part we present the GeoGebra interface, the main tools used in the construction of polyhedron representations and the use of GeoGebra in the teaching of geometry, especially in the teaching of polyhedrons. From the constructions carried out with GeoGebra, we concluded that the GeoGebra software promises a satisfactory construction of the polyhedron classes studied for the teaching outcome in the teaching of this content.