This is an overview of geometric algebras in dimensions 0-4 from the perspective of the concept of number itself. It is developed from a historic viewpoint and investigates and develops a pedagogic approach emphasizing the geometric aspects of the subject. There is a focus broadly on three main but interconnected areas: the relation between the discrete and the continuous, the centrality of complex numbers, and the hypothesis that the octonions may be expressed in the even subalgebra of four dimensions. This will not be proved, and the focus is on the overall perspective and presentation. One central result is a proposal for the identification of the Cayley-Dickson process with the gluing together of spheres, extending it to start from ℕ. This leads to a projective representation of a unified elliptic/parabolic/hyperbolic geometry of 4-dimensional space. Coupled with this is a discrete representation of it that can be classified with Geometric Algebra.
| Identifer | oai:union.ndltd.org:UPSALLA1/oai:DiVA.org:uu-507809 |
| Date | January 2023 |
| Creators | Gunnarsson, Petter |
| Publisher | Uppsala universitet, Analys och partiella differentialekvationer |
| Source Sets | DiVA Archive at Upsalla University |
| Language | English |
| Detected Language | English |
| Type | Student thesis, info:eu-repo/semantics/bachelorThesis, text |
| Format | application/pdf |
| Rights | info:eu-repo/semantics/openAccess |
| Relation | U.U.D.M. project report ; 2023:28 |
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