Tetrahedra tiling problem

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.