## Publication: Rational trigonometry of a tetrahedron over a general metrical framework

 dc.contributor.advisor Wildberger, Norman J. en_US dc.contributor.author Notowidigdo, Gennady en_US dc.date.accessioned 2022-03-23T09:21:37Z dc.date.available 2022-03-23T09:21:37Z dc.date.issued 2018 en_US dc.description.abstract This thesis sets up a framework for rational trigonometry in three dimensions, using a linear algebraic approach to extend the classical trigonometric framework of years past, as well as the two-dimensional rational trigonometric framework of Wildberger, beyond the usual Euclidean setting to arbitrary symmetric bilinear forms and arbitrary fields not of characteristic 2. We will use two complementary techniques to establish such a framework. In addition to a generalised scalar product which is defined by a symmetric bilinear form, we define a generalised vector product. Furthermore, we derive analogs of classical results attributed to Lagrange, Cauchy and Binet, and use these to establish formulas for the quadrances, quadreas, quadrume, spreads, dihedral spreads, solid spreads and dual solid spreads of a general tetrahedron. While we aim to generalise and prove previously stated formulas of Wildberger, as well as classical formulas attributed to Richardson, we also establish new results such as the Three-dimensional quadrea theorem and the Quadrume theorem. The other technique is to introduce standard co-ordinates, where affine transformations are used to transform to a particularly simple example, and all the complexity resides in the algebraic expression for the symmetric bilinear form rather than the generality of the tetrahedron itself. Using this technique, we derive the Tetrahedron cross law and the Dihedral cross relation. Throughout this thesis, we use a simple example from Khafre’s pyramid to illustrate the ideas we have formulated, and in the final chapter we examine the special cases of the regular, isosceles and trirectangular tetrahedral, as well as a general tetrahedron in a relativistic setting and a general tetrahedron over a finite field. en_US dc.identifier.uri http://hdl.handle.net/1959.4/61277 dc.language English dc.language.iso EN en_US dc.publisher UNSW, Sydney en_US dc.rights CC BY-NC-ND 3.0 en_US dc.rights.uri https://creativecommons.org/licenses/by-nc-nd/3.0/au/ en_US dc.subject.other Linear algebra en_US dc.subject.other Trigonometry en_US dc.subject.other Geometry en_US dc.subject.other Rational trigonometry en_US dc.subject.other Tetrahedron en_US dc.subject.other Affine geometry en_US dc.subject.other Projective geometry en_US dc.title Rational trigonometry of a tetrahedron over a general metrical framework en_US dc.type Thesis en_US dcterms.accessRights open access dcterms.rightsHolder Notowidigdo, Gennady dspace.entity.type Publication en_US unsw.accessRights.uri https://purl.org/coar/access_right/c_abf2 unsw.identifier.doi https://doi.org/10.26190/unsworks/20973 unsw.relation.faculty Science unsw.relation.originalPublicationAffiliation Notowidigdo, Gennady, Mathematics & Statistics, Faculty of Science, UNSW en_US unsw.relation.originalPublicationAffiliation Wildberger, Norman J., Mathematics & Statistics, Faculty of Science, UNSW en_US unsw.relation.school School of Mathematics & Statistics * unsw.thesis.degreetype PhD Doctorate en_US
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