Recently I began looking into tet-meshes again. Some colleagues and I were trying to wrap our heads around how many tetrahedra result from splitting up a triangular prism. we were trying to draw out the cases on the white board. Occasionally some one would say, “I’m sure it’s 3.” Then somebody else would draw another set of pictures and’d say “No, I’m sure it’s 4″.
Well, to have a definitive answer, a triangular prism may be split into three tetrahedra. If you don’t believe me, see the nice description (which actually lists all possible splits) in “How to Subdivide Pyramids, Prisms and Hexahedra into Tetrahedra” by J. Dompierre et al. in 1999. There is another interesting discussion in “Space-filling Tetrahedra in Euclidean Space” by D.M.Y. Sommerville in 1923, in which they consider the conditions under which these tetrahedra are congruent.