• Algebra and Geometry of Hamilton's Quaternions: 'Well, Papa, Can You Multiply Triplets?'

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      https://www.ias.ac.in/article/fulltext/reso/021/06/0529-0544

    • Keywords

       

      Quaternions, division algebra, rotations, spheres, Hopf map.

    • Abstract

       

      Inspired by the relation between the algebra ofcomplex numbers and plane geometry, WilliamRowan Hamilton sought an algebra of triples forapplication to three-dimensional geometry. Unableto multiply and divide triples, he inventeda non-commutative division algebra of quadruples,in what he considered his most significantwork, generalizing the real and complex numbersystems. We give a motivated introduction toquaternions and discuss how they are related toPauli matrices, rotations in three dimensions, thethree sphere, the group SU(2) and the celebratedHopf fibrations.

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