Tetrahedra tiling problem

TL;DR

This paper investigates which classified tetrahedra can tile space, revealing that only specific families and sporadic cases do, and disproving a related conjecture about Dehn invariant zero tetrahedra.

Contribution

It determines the space-tiling capability of classified tetrahedra families, providing new insights into their geometric properties and tiling potential.

Findings

Every Hill family member tiles space

Exactly one new family member tiles space

At most 40 sporadic tetrahedra tile space

Abstract

Kedlaya, Kolpakov, Poonen, and Rubinstein classified tetrahedra all of whose dihedral angles are rational multiples of $\pi$ into two one-parameter families (a Hill family and a new family) and $59$ sporadic tetrahedra. In this paper, we consider which of them tile space; we show that every member of the Hill family, exactly one member of the new family, and at most $40$ sporadic tetrahedra tile space. As a corollary, we disprove the converse of Debrunner's theorem, showing that not all Dehn invariant zero tetrahedra tile space.