Poliedros regulares e semirregulares

Abstract

In this Undergraduate Final Project we present a study of the five regular polyhedra (or Platonic Solids) and the thirteen semi-regular polyhedra (or Archimedean Solids). This study was divided into two parts: theoretical part, with deductions from formulas and classification of above polyhedra and; practical part, with the dynamic construction of semi-regular polyhedra from regular polyhedra using the GeoGebra software. It is important to emphasize the impossibility of dynamically constructing semi-regular polyhedra in GeoGebra without some of the formulas related to regular polyhedra that were deduced in the theoretical part. The formulas deduced were: (i) Euler Relation for convex polyhedra; (ii) measurements of the central and dihedral angles of a regular polyhedron depending on its genus of faces and its genus of vertices; (iii) radii of spheres inscribed and circumscribed by a regular polyhedron depending on the genus of faces, the genus of vertices and the length of the edges; (iv) apothem and radius of the circle circumscribed to the face of a regular polyhedron depending on its genus of faces and its edge length and; (v) area and volume of a regular polyhedron as a function of its genus of faces, its genus of vertices, its number of faces and its edge length. It is also important to highlight that the constructions of semi-regular polyhedra were made through three geometric operations on regular polyhedra: (1) simple truncation; (2) composite truncation and; (3) snubification. In this work we hope to be able to contribute to the theory of regular and semi-regular polyhedra, as well as the dynamic geometric constructions of such polyhedra in the GeoGebra software