Compass and Straightedge Constructions in the Hyperbolic Plane
Summary
At the dawn of mathematics, the straightedge and compass were important tools for classical geometers. A prime example are The Elements of Euclid, which cover a substantial portion of Greek geometry and has been a source of study for generations to come. Up until the 19th century, the geometrical world shaped by Euclid has been the sole focus of the field, when by a slight adjustment in one of the postulates an entirely new theory was born.
This thesis will take the classical approach towards the hyperbolic plane, as we explore straightedge and compass constructions in this space. At last, this inquiry will lead us to necessary and sufficient conditions for line segments and angles to be constructible. As a consequence, it will be demonstrated that some squares and circles of equal area are constructible in the Hyperbolic plane, whereas this is a notorious impossibility in Euclidean space, and precise conditions will be provided.
Furthermore, the straightedge and compass are studied shortly in the Euclidean plane, as well as Hilbert's axioms, which offer a rigorous foundation to synthetic geometry.