The values for the tetrahedron, cube, octahedron, dodecahedron and icosahedron are:3.27, 3.27, 3.46,3.19 and 3.64, resp. These produce the largest d for N=4,8,

The values for the tetrahedron, cube, octahedron, dodecahedron and icosahedron are:3.27, 3.27, 3.46,3.19 and 3.64, resp. These produce the largest d for N=4,8,6,12 and 20 resp. Thus, it is presumed by almost everyone that d=3.64 is the global upper bound. Unfortunately, I believe it is still an open problem to to prove this or to describe a general upper bound for specific N not equal to any of these five values.