Construções geométricas: possíveis e impossíveis

Abstract

The present work proposes to explore the diverse geometric constructions made with drawing instruments, from those that are possible, the approximate and even the impossible, made with the ruler and the compass. Starting from a historical exploration of the main constructions that allowed mathematics to develop in ancient civilizations, as well as those that took centuries to be demonstrated as impossible, causing mathematicians to create different approaches to Euclidean geometry. In particular, there are three problems with impossible geometric constructions and even today they are known as “The Three Classical Problems of Antiquity”: squaring the circle, angle trisection and cube duplication. In this text, the paths that led to the discovery of the irresolubility of these problems were exposed, from the algebraization of geometric constructions, treating each step as an operation between the segments and primitive elements of Euclidean geometry, to the resources developed later that allowed to obtain a solution of such problems. Finally, constructive numbers were also discussed, starting from the connection between geometric constructions and basic mathematical operations, allowing mathematicians to generate approximate constructions of various geometric objects and segments of lengths impossible to be generated perfectly with the basic construction tools.