Determining all universal tilers (Q1943656)

The intersection \(C\) of a plane and a convex polyhedron is called a cross-section if \(C\) is a polygon with at least 3 edges. A universal tiler is defined as a convex polyhedron whose every cross-section tiles the plane. The tilings are not necessarily edge-to-edge. The main result is the following {Theorem}. A convex polyhedron is a universal tiler if and only if it is a tetrahedron or a pentahedron with parallel facets. A graphical illustration is given in the conclusion. If the universal tiler is a pentahedron \(P\) then it has two triangular bases, and either \(P\) is a triangular prism, or all the quadrilateral faces of \(P\) are trapezoids.