Triangulations of spheres and discs

Abstract

The main ob jective of this research is to find the different types of elliptic triangulations for planar discs and spheres. We begin in Chapter 1 with the mandatory introduction. In the second chapter we define and study the notion of a patch, that is, a triangulation of a planar disc. By introducing a suitable notion of degree for the vertex points, we focus on those patches with points of degrees less tah or equal to 6. Such patches are called el liptic. We show that the elliptic patches with precisely three points of degree four, denoted by (0, 3, 0), can be classified. The number of vertex points of these patches are calculated, and we also describe their triangulation structures. From the classification of patches of the type (0, 3, 0), we describe and find the number of vertex points for three other elliptic patches of types (0, 2, 2), (0, 1, 4), (0, 0, 6). We also describe an enlargement method for constructing patches (which we call the generic construction method) and apply this method to derive formulas for certain patches. In the third chapter we describe some configurations for constructing elliptic spherical triangulations. These are the mutant configu- ration, the productive configuration and the self-reproductive configuration. We also describe the face-fullering and edge-fullering methods as well as the glueing of patches method for constructing triangulations and patches. We show that there are only 19 possible types of elliptic triangulations for spheres and determine the existence (as well as nonexistence) of all these types except for a small number of cases.