On nontriangulable polyhedra

Abstract

Triangulations of 3-dimensional polyhedron are partitions of the polyhedron with tetrahedra in a face-to-face fashion without introducing new vertices. Schonhardt (Math. Ann. 89:309–312, 1927), Bagemihl (Amer. Math. Mon. 55:411–413, 1948), Kuperberg (Personal communication 2011) and others constructed special polyhedra in such a way that clever one line geometric reasons imply nontriangulability. Rambau (Comb. Comput. Geom. 52:501–516, 2005) proved that twisted prisms over n-gons are nontriangulable. Our approach for proving polyhedra are nontriangulable is to show that partitions with tetrahedra, which we call tilings, do not exist even if the face-to-face-restriction is relaxed. First we construct a polyhedron which is tileable but is not triangulable. Then we revisit Rambau type twisted prisms. In fact we consider a slightly different class of polyhedra, and prove that these new twisted prisms are nontileable, thus are nontriangulable. We also show that one can twist the regular dodecahedron so that it becomes nontileable, which is abstracted to a new family of nontileable polyhedra, called nonconvex twisted pentaprisms.