Tiling Polyhedra with Tetrahedra

When solving an algorithmic problem involving a polyhedron in R, it is common to start by partitioning the given polyhedron into simplier ones. The most common process is called triangulation and it refers to partitioning a polyhedron into tetrahedra in a face-to-face manner. In this paper instead of triangulations we will consider tilings by tetrahedra. In a tiling the tetrahedra are not required to be attached to each other along common faces. We will construct several polyhedra which can not be triangulated but can be tiled by tetrahedra. We will also revisit a nontriangulatable polyhedron of Rambau and a give a new proof for his theorem. Finally we will identify new families of non-tilable, and thus non-triangulable polyhedra.