On universal tilers

A famous problem in discrete geometry is to find all monohedral plane tilers. It is still open to the best of our knowledge. This paper is concerned with one of its variants, that of determining all convex polyhedra whose every cross-section tiles the plane. We call such polyhedra universal tilers. We obtain that a convex polyhedron is a universal tiler only if it is a tetrahedron or a pentahedron.