Sommerville's missing tetrahedra (Q878075)

In 1923 \textit{D. M. Y. Sommerville} [Edinb. R. S. Proc. 43, 85--116 (1923; JFM 49.0411.03)] determined a list of four tetrahedra such that there exist face-to-face tilings of the Euclidean space by properly congruent copies of a tetrahedron. The paper under review considers the case which was not examined by Sommerville. It is proved that the missing tetrahedron does not admit any face-to-face tiling of the Euclidean space if the copies of tetrahedron are properly congruent, i.e. congruent by an orientation-preserving isometry. Therefore, the Sommerville's classification is complete. However this tetrahedron admits tilings that are either non-proper face-to-face or proper non-face-to-face. Remark that no assumptions about a symmetry group of the tiling are made.