Equidecomposição de figuras e o terceiro problema de Hilbert

Abstract

This paper aims to show the solution given by Max Dehn to the third Hilbert’s problem, which is the following question “there are two fi gures in space that are not equivalent by dissection and have the same volume?”. The work is divided into three chapters: in the fi rst presents a brief introduction on units of measure both lengths and areas. A function is constructed which measures the part of the plane occupied by afigure, it is called area function. The second chapter is devoted to the study of poly-gons that are equivalent by dissection, i.e., polygons that may be constructed splitting one of then in smaller parts that do not overlap and, when placed together in another position, form the other polygon. It also a proof of the Bolyai-Gerwien theorem which establishing the area as an invariant for equivalence by dissection. The third chapter deals with equivalence by dissection in space, with a discussion of Proposition 5 of Book XII of “The Elements” of Euclid and the third problem of Hilbert, and the solution given by Max Dehn to the Hibert’s third problem is presented. One important feature of Max Dehn proof is the use of elements of an abelian group as invariants to equivalence by dissection. A consequence of Dehn’s theorem is that the principle of exhaustion is necessary in the study the volume of pyramids.