Universal Hyperbolic Geometry, Sydpoints and Finite Fields: A Projective and Algebraic Alternative

Universal hyperbolic geometry gives a purely algebraic approach to the subject that connects naturally with Einstein's special theory of relativity. In this paper, we give an overview of some aspects of this theory relating to triangle geometry and in particular the remarkable new analogues of midpoints called sydpoints. We also discuss how the generality allows us to consider hyperbolic geometry over general fields, in particular over finite fields.