Acute Triangulations Of Two Archimedean Polyhedron 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)Ï€2.In this thesis we consider the non-obtuse and acute triangulations of the surfaces ofthe Archimedean solids icosidodecahedron and snub cube, and prove that the surface oficosidodecahedron can be triangulated into 8 acute (resp. non-obtuse) triangles, and 8 isthe best possible. Furthermore, we prove that the surface of snub cube can be triangulatedinto 8 non-obtuse triangles and 12 acute triangles, and both of the bounds 8 and 12 arethe best possible.