Foundations of Hyperbolic Geometry

In recent work by Bamberg and Penttila (2023), Helen Skala’s (1992) elegant firstorder axiom system for plane hyperbolic geometry has been further simplified with the replacement of two assumptions with two simpler axioms, the independence of the second of which, the “perpendicularity axiom”, remains in question. This project aims to determine whether the perpendicularity axiom is necessary for a complete and independent set of axioms for plane hyperbolic geometry. The methodology is to use an algebraic form of the axioms and search for a proof of the perpendicularity axiom by codifying the symbolic statements in an interactive proof solver. The geometric interpretation of all results will be considered, and findings will be visualised.

Jolyon Joyce

The University of Western Australia

Jolyon is a Bachelor of Philosophy (Honours) student at the University of Western Australia, studying a double major in mathematics and physics. In 2021, he undertook an undergraduate research placement in theoretical physics with Dr. Darren Grasso, culminating in a derivation of the action for electromagnetic fields from Maxwell’s
equations. He completed a semester on exchange at the University of Manchester in the second half of 2022, where he was introduced to the field of mathematical logic. His focus is now on pure mathematics, with a particular interest in logic, geometry, and algebra. In his spare time, he enjoys making music: playing piano, singing in choirs, and occasionally composing. Since 2022 he has been President of The Winthrop Singers Ltd, a tertiary student choir and not-for-profit based in Perth. He looks forward to commencing Honours in Pure Mathematics at UWA in 2024.

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