Completeness of the classification of face-to-face tetrahedral space-fillers

Determine whether every tetrahedral space-filler arises by partitioning a triangular prism or a parallelohedron, thereby resolving the completeness question for the Sommerville and Goldberg lists of face-to-face tetrahedral tiles.

Background

The addendum discusses the tetragonal disphenoid T as one of the classical tetrahedra that tile Euclidean three-space. It cites a conjecture that the known Sommerville and Goldberg classifications of face-to-face tetrahedral tiles are complete and notes the related necessary condition that every tetrahedral tile might have a right dihedral angle.

The unresolved structural question is whether all tetrahedral space-fillers can be generated through the two indicated construction classes: partitions of triangular prisms or partitions of parallelohedra.

References

Conjecture~11.2 of : the Sommerville and Goldberg lists of face-to-face tetrahedral tiles are complete, and in particular every tetrahedral tile has at least one dihedral angle equal to $\pi/2$. $T$ has two. It remains open whether every tetrahedral space-filler arises by partitioning a triangular prism p.~242, or a parallelohedron.

— Schwarz $\mathrm{P}$ and Schwarz $\mathrm{D}$ as free-boundary minimal disks in a tetragonal disphenoid: exact areas  (2610.22013 - Finch, 15 Oct 2026) in Section 9, Addendum A, subsection “Three open problems that touch $T$,” item (a)