O Teorema de Euler para poliedros e aplicações

Abstract

The geometry is one of the fundamental areas of mathematics, playing a central role in the understanding of shapes and spatial structures. Among the most relevant geometric objects in this context are polyhedra, whose surfaces are composed of a finite number of planar faces, edges, and vertices. One of the most important relations associated with these solids is Euler’s Theorem, which establishes an invariant connection between the number of vertices, edges, and faces of a convex polyhedron. Euler’s Theorem for convex polyhedra is examined from a geometric, combinatorial, and topological perspective, with emphasis on its interpretation as a topological invariant. Fundamental concepts of geometry, surfaces, and polyhedra are considered, as well as different classical proofs of the theorem, highlighting the diversity of approaches employed in its historical and mathematical foundation. The Euler relation is also analyzed in the plane, establishing connections with planar graphs and leading to the concept of the Euler characteristic, which extends the result to more general surfaces. In addition, applications are explored in different contexts, such as Platonic solids, graph theory, computer graphics, engineering, architecture, molecular structures, and cartography, emphasizing the relevance of Euler’s Theorem both in mathematics and in interdisciplinary areas.