Poliedros de kepler-Poinsot: explorando poliedros regulares não convexos com o geogebra

Abstract

This work deals with regular non-convex polyhedra, known as Kepler-Poinsot polyhedra. Its general objective is to use dynamic geometry resources to expand the understanding about the following questions: why are Kepler-Poinsot polyhedra considered regular polyhedra? What are the vertices, edges and faces of these polyhedra? Although these questions may seem obvious, answering them requires diving into another question: what, in fact, is a polyhedra? During the development of the work, we showed that there is more than one way to define what a polyhedron is and we sought, with the help of the Geogebra software, to make it clear which polygons should be understood as faces of Kepler-Poinsot polyhedra so that they can be interpreted as regular polyhedra. Considering the different possibilities in choosing a definition of a polyhedron suitable for our purposes, we opted for the one proposed by Branko Grünbaum (2003), which differentiates the combinatorial structure of a polyhedron from its geometric interpretation, allowing a more general understanding of the structures we call a polyhedra. According to the definition we chose, a face is understood as a circuit composed of vertices and edges, in such a way that the faces that constitute the Kepler-Poinsot polyhedra can be interpreted as intersecting regular polygons. Answering all these questions requires a lot of abstract thinking, which justifies the use of dynamic geometry as a visual support in order to facilitate the path that leads to such abstractions. Thus, the exposition made throughout this work makes use of constructions produced by the author and made available on the Geogebra online platform to make these ideas clearer, thus achieving the proposed objectives.