On Universal Tilers

TL;DR

This paper investigates convex polyhedra whose every cross-section can tile the plane, establishing that only tetrahedra and pentahedra possess this property, thus advancing understanding in discrete geometry.

Contribution

It characterizes all convex polyhedra that are universal tilers, showing only tetrahedra and pentahedra meet this criterion, which is a new classification result.

Findings

Only tetrahedra and pentahedra are universal tilers.

Cross-sections of these polyhedra can tile the plane.

Provides a complete classification of such polyhedra.

Abstract

A famous problem in discrete geometry is to find all monohedral plane tilers, which is still open to the best of our knowledge. This paper concerns with one of its variants that to determine 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.