Schönhardt polyhedron

The Schönhardt polyhedron is the simplest polyhedron that cannot be triangulated into tetrahedra. It is a concave variant of the triangular gyroprism.

It cannot be divided into (finitely many) tetrahedra without introducing new vertices. Rambau showed that this is also true of every "nonconvex twisted prism", or concave gyroprism, where the base triangle is replaced by another non-degenerate regular polygon.