Algebra and geometry of Hamilton's quaternions

TL;DR

This paper introduces Hamilton's quaternions, exploring their algebraic structure and geometric applications, including rotations, the three sphere, and connections to Pauli matrices and SU(2).

Contribution

It provides a motivated introduction to quaternions and details their relation to key mathematical and physical concepts, highlighting their significance in three-dimensional geometry.

Findings

Quaternions form a non-commutative division algebra.

They are related to Pauli matrices and SU(2).

Quaternions describe rotations in three dimensions.

Abstract

Inspired by the relation between the algebra of complex numbers and plane geometry, William Rowan Hamilton sought an algebra of triples for application to three dimensional geometry. Unable to multiply and divide triples, he invented a non-commutative division algebra of quadruples, in what he considered his most significant work, generalizing the real and complex number systems. We give a motivated introduction to quaternions and discuss how they are related to Pauli matrices, rotations in three dimensions, the three sphere, the group SU(2) and the celebrated Hopf fibrations.