Rational trigonometry of a tetrahedron over a general metrical framework

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.