Acute Triangulations Of Polyhedral Surfaces

A triangulation of a two-dimensional space means a collection of (full) trianglescovering the space, such that the intersection of any two triangles is either empty orconsists of a vertex or of an edge. A triangle is called geodesic if all its edges are seg-ments, i.e., shortest paths between the corresponding vertices. We are interested only ingeodesic triangulations, all the members of which are, by definition, geodesic triangles.An acute (resp. non-obtuse) triangulation is a triangulation whose triangles have all theirangles less than (resp. not greater than) 90?.In this thesis we firstly settle an open problem on the acute triangulations of theregular dodecahedral surface, and prove that the surface of the regular dodecahedronadmits an acute triangulation with 12 triangles, which is the best possible. Then weconsider the non-obtuse and acute triangulations of the the surfaces of the Archimedeansolids rhombicuboctahedron and truncated octahedron. We prove that the surface ofthe rhombicuboctahedron can be triangulated into 8 non-obtuse triangles and 12 acutetriangles, where both of the bounds are the best possible; the surface of the truncatedoctahedron can be triangulated into 8 non-obtuse triangles and 12 acute triangles, where8 and 12 are also the best possible bounds.