sy to be solved analytically and, as far as I know, there is no closed form solution for a generic number of points. A numerical solution for M = 4, 8, and 16 is presented in [37] where both the angular coordinates indicating the position of the points on the Poincarè sphere and the normalized distance matrixes are reported. The optimum constellations are in a certain sense regular, but do not reproduce the vertices of polyhedrons or other known tridimensional forms. On the other hand, in [36] regular constellations are proposed for M-PolSK. Considering the vertices of regular polyhedron in the space, the possible M-PolSK for- mats the minimum distance in the Stokes space and the minimum distance for the opti- mum configuration with the same number of points are listed in Table 9.3. TABLE 9.3 Minimum Normalized Distance in the Stokes Space for the Configurations Derived from Regular Polyhedron in the Stokes Space and the Minimum Distance for the Optimum Configuration with the Same Number of Points Minimum PolSK Symbols # Regular Solid Distance, d Optimum, dopt Deviation (d - dopt)% Tetrahedron 4/J6 = 1.633 1.633 Octahedron V2 = 1.414 2/V3 = 1.155 Icosahedron Dodecahedron Snub dodecahedron 0.464 Icosidodecahedron =0.263 There are four Archimedean solids with 60 vertices and the snub Dodecahedron is the best under a mini- mum distance point of view. Telecommunication Networks Icosidodecahedron FIGURE 9.25 Two of the Archimedean solids used to build high order M-PolSK constellations. In the figure, both the plane development and a per- spective three dimensional draw- ing are represented. Snub dodecahedron To construct regular constellations, Pythagorean and Archimedean solids are used as appropriates to achieve a given number of symbols. When several such solids exist with the same number of vertices, the best from the minimum distance point of view is selected [38,39]. Two of the less known regular solids cited in Table 9.3 are represented both in plane development and in tridimensional representation in Figure 9.