Investigation of inversion in some non-euclidean geometries induced by convex polyhedra

In this thesis inversions have been studied in two Minkowski type geometries whose spheres are disdyakis dodecahedron which is a catalan solid and cuboctahedron which is an Archimedean solid. In that geometries inversions have been defined and investigated firstly with respect to a circle by reducing metrics with the projection method in the plane, after that inversions defined and investigated with respect to a sphere in the space. In the first chapter of this study, convex polyhedra, the relations between convex polyhedra and Minkowski Geometries and inversions are briefly mentioned. In the second chapter definitions metric, inversions, cross ratio and harmonic conjugates have been given. In the third chapter inversion with respect to a circle in the maximum plane and inversion with respect to a circle in the disdyakis dodecahedron plane, respectively, have been given with subsections, theorems and proofs. Finally in the fourth chapter inversions with respect to a sphere in the Cuboctahedron and the Disdyakis Dodecahedron spaces are studied and cross ratio and harmonic conjugates in this spaces are investigated.