On the analogues of the Pythagorean and Thales theorems in some non-Euclidean geometries

This thesis focuses on defining the counterparts of Minkowski-type geometries in the plane, which are constructed with metrics previously defined in three-dimensional analytical space, considering certain convex polyhedra as spheres. Additionally, the thesis investigates the analogues of Thales' and Pythagoras' theorems in these planes. The thesis consists of seven chapters. In the first chapter, a literature review is conducted to highlight the importance of the problem addressed in the thesis and to outline the sources of motivation. The second chapter presents some essential definitions and theorems used throughout the thesis. Chapters three, four, five, and six constitute the original contributions of the thesis. These chapters introduce and define the planes of the Disdyakis Dodecahedron, Triakis Octahedron, Truncated Octahedron, and Tetrakis Hexahedron, respectively. After providing their properties, the study examines the analogues of Thales' and Pythagoras' theorems in these planes. Finally, the seventh chapter summarizes the results obtained, discusses their contribution to the literature, and provides recommendations for future studies based on the work carried out in this thesis.