The Hamilton-Cayley formula

• Grant number: 15/15219-3
• Support Opportunities: Scholarships in Brazil - Scientific Initiation
• Start date: October 01, 2015
• End date: September 30, 2016
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Jaime Edmundo Apaza Rodriguez
• Grantee: Karina Aparecida da Silva
• Host Institution: Faculdade de Engenharia (FEIS). Universidade Estadual Paulista (UNESP). Campus de Ilha Solteira. Ilha Solteira , SP, Brazil

Abstract

This project's overall objective is to show that there is a strong interaction between Geometry and algebra, using the language of quaternions. In general, in the curricula of undergraduate courses in mathematics, the Abstract Algebra disciplines (rings, fields and Groups) and the geometry disciplines (including Differential Geometry) are presented in way "isolated" without being given their interrelationships. Quaternions are defined and studied for a gradual enrichment of mathematical structure: The Cartesian product of quaternions sets consecutively are transformed into a real vector space and a 4D Algebra associative division, not commutative, over the real numbers and identity element. A relation of the quaternions with Euclidean Geometry 4D is established. The Euclidean space 4D is presented with Cartesian coordinates, in order to apply the matrix algebra. The main results are a representation of the quaternion rotations4D and a proof of the Hamilton-Cayley formula for 3D rotations.