Dehn invariant zero tetrahedra

TL;DR

This paper surveys known Dehn invariant zero tetrahedra, introduces two new families, and explores the finiteness of such tetrahedra with certain angle span conditions, advancing classification efforts.

Contribution

It provides a comprehensive survey, discovers two new families, and investigates the finiteness of tetrahedra with five-dimensional angle span over rationals.

Findings

Identified two new families of Dehn invariant zero tetrahedra.

Proved finiteness of such tetrahedra with five-dimensional angle span.

Progressed towards classifying all Dehn invariant zero tetrahedra.

Abstract

We survey literature on all known families and examples of Dehn invariant zero tetrahedra. We also contribute two previously unknown families of Dehn invariant zero tetrahedra. Following a suggestion of Dill and Habegger, we show that there are finitely many Dehn invariant zero tetrahedra whose dihedral angles have a five-dimensional span over $\mathbb{Q}$, a first step towards classifying all Dehn invariant zero tetrahedra.