Determining All Universal Tilers

TL;DR

This paper characterizes all convex polyhedra that can produce plane-tiling cross-sections, proving only tetrahedra and triangular prisms qualify as universal tilers through a novel cross-section rotation method.

Contribution

It introduces a new slight-rotating operation for cross-sections and provides a complete classification of convex polyhedra that are universal tilers.

Findings

Only tetrahedra and triangular prisms are universal tilers.

The slight-rotating operation is key to the classification.

The proof relies on analyzing cross-section tiling properties.

Abstract

A universal tiler is a convex polyhedron whose every cross-section tiles the plane. In this paper, we introduce a certain slight-rotating operation for cross-sections of pentahedra. Based on a selected initial cross-section and by applying the slight-rotating operation suitably, we prove that a convex polyhedron is a universal tiler if and only if it is a tetrahedron or a triangular prism.