On a Generalization of Schönhardt's Polyhedron

Summary

We show that the nonconvex twisted prism over an n-gon cannot be triangulated without new vertices. For this, it does not matter what the coordinates of the n-gon are as long as the top and the bottom n-gon are congruent and the twist is not too large. This generalizes Schönhardt's polyhedron, which is the nonconvex twisted prism over a triangle.

1. The Background

Lennes [1911] was the first to present a simple three-dimensional nonconvex polyhedron whose interior cannot be triangulated without new vertices. The more famous example, however, was given by Schönhardt [1927]: he observed that in the nonconvex twisted triangular prism (subsequently called “Schönhardt's polyhedron“) every diagonal that is not a boundary edge lies completely in the exterior. This implies that there can be no triangulation of it without new vertices because there is simply no interior tetrahedron: all possible tetrahedra spanned by four of its six vertices would introduce new edges. Moreover, he proved that every simple polyhedron with the same properties must have at least six vertices. Later, further such nonconvex, nontriangulable polyhedra with an arbitrary number of points have been presented. Among them, Bagemihl's polyhedron [1948] also has the feature that every nonfacial diagonal is in the exterior.

The nonconvex twisted prism over an arbitrary n-gon would arguably be the most natural generalization of Schönhardt's polyhedron. Surprisingly enough, there has been no proof so far that it cannot be triangulated without new vertices. One of the reasons seems to be that — in contrast to Schönhardt's and Bagemihl's polyhedra — not every nonfacial diagonal lies completely outside the polygonal prism.