Nice question! Cleverly avoiding addressing it, I wonder: * If there is a good way of ordering the f-vectors F = (f_0, f_1, f2, f_3) (where f_k is the count of k-dimensional faces of a polyhedron, so Fred's dissection would have F = (20, 61, 66, 24)) so that one could seek a polyhedron for which F is optimal with respect to this ordering. * I've seen similar questions for polygons where the dissection sought is one into acute triangles. E.g., the minimal number of these for a square is 7. So: What is the smallest number of acute tetrahedra that a regular solid can be dissected into? An acute tetrahedron is one whose dihedral angles are all acute.* —Dan _______________ P.S. Just learned two things on the subject: 1) The smallest known number of acute tetrahedra that the cube can be dissected into (as of 2010) was 1370. 2) It is known that 3-space can be dissected into acute simplices, but 5-space cannot be. ----- Fred <fred.lunnon@gmail.com> wrote on Dec 12, 2016 at 10:48 AM: ----- Problem: dissect into tetrahedra the regular dodecahedron with interior, so that the resulting complex is symmetric under reflection in its centre. I have a solution with 20 vertices, 61 edges, 66 triangles, 24 tetrahedra: can any of these counts be reduced? [ Further details concerning this solution, and the motivation behind the problem, available on demand. ] -----