On the geometry of dodecahedron metric

This thesis has been studied on some geometric structures and their properties in the plane and space furnished with the dodecahedron metric. This thesis consists of six sections. The first section is the introduction section and in this section, a literature review is given on the studies that shed light on the formation of the subject of the thesis. In the second section, some basic definitions and theorems related to the subject to be studied are briefly mentioned. In the third section, the dodecahedron metric in the plane is obtained by projection from its defined state in the 3-dimensional analytic space, some properties of this metric are mentioned and conics are examined in the plane furnished with this metric by considering their special cases. In the fourth section, inversions with respect to the circle and ellipse in the dodecahedron plane and inversion with respect to the sphere in the dodecahedron space are defined and the basic concepts of these transformations are given and some properties are mentioned. In the fifth section, inversions with respect to the circle, ellipse and sphere and the relations of the concepts of cross ratio and harmonic conjugate are studied. Finally, in the sixth chapter, the results obtained from the subject studied in the thesis are summarized and the studies that can be done with the help of these are mentioned.