Clifford e as etapas da elaboração das álgebras geométricas

Abstract

This dissertation studies the steps of the elaboration of geometric algebras in Clifford's texts. The first chapter addresses Clifford's training and his first articles, with the aim of understanding which subjects he was interested in the first years of his career. It appears that projective geometry is present in most texts, and Clifford was greatly influenced by Salmon's works and Cayley's memories. The second chapter studies the Preliminary Sketch of Biquaternions (CLIFFORD, 1873), which is the first article in which Clifford talks about rotors, motors and biquaternions, in an attempt to generalize Hamilton's quaternions by creating a theory that accounts for the movements of a rigid body in space. This article represents an initial milestone in this theory and the heart of this dissertation. Clifford also discusses this theory in elliptical space. In the fourth chapter, we see how the generalization of geometric algebras happened in Clifford's texts after 1873. Clifford not only unifies Grassmann's geometric product and Hamilton's quaternions into a single structure, but generalizes it in n dimensions. At this point, Clifford considers the possibility that the squares of imaginary units are equal to +1, not -1, differently from how it is done in complex numbers and quaternions, an approach that is used in modern geometric algebra texts.