Extremal Properties of Regular Polyhedra

On the surface of a ball the problems of the densest circle packing and the thinnest circle covering with 4, 6, or 12 congruent circles lead to arrangements of circles in which their centers are vertices of a regular tetrahedron, octahedron, or icosahedron. In contrast, in the most favorable arrangement of 8 or 20 circles, their centers are not the vertices of a regular hexahedron or dodecahedron. Hence for these problems, the regular triangular polyhedra play a special role. However, since for example in the densest packing of 12 congruent circles on the sphere, their planes bound a regular dodecahedron, the same problems can be formulated so that the regular three-valent polyhedra play a special role.